IRR Calculator

Find r such that −initial investment + CF₁ ÷ (1+r) + CF₂ ÷ (1+r)² + … + CFₙ ÷ (1+r)ⁿ = 0. There is no algebraic solution for r when there are more than a couple of periods, so it is found by narrowing an interval until the net present value is zero to two decimal places.

Initial investment

What the project costs you today, at the moment you start. It is entered as a positive number and subtracted in the equation rather than written as a negative cash flow, which keeps the sign convention visible instead of buried in the list.

Cash flows

What comes back, period by period, separated by spaces. The first entry is one period from now, not now — money received on day zero belongs in the initial investment as a reduction. The periods are whatever you want them to be, months or years, as long as the discount rate is expressed over the same one.

Discount rate

Your hurdle rate for this page: what money costs you, or the return you could get elsewhere at the same risk. It does not change the internal rate of return at all. It decides whether the project clears the bar, so it is the number that turns a rate into an answer.

Net present value

What the project is worth today once every future receipt is discounted at your rate and the cost is subtracted. Positive means it beats the hurdle; the internal rate of return is the rate at which this figure crosses zero.

Internal rate of return

The discount rate that makes the net present value zero, expressed as a percentage. It is a property of the cash flows alone — the same project has the same internal rate of return whatever hurdle you happen to be using — which is exactly why the two outputs can disagree about nothing and still look like they are giving different advice.

Use it when you want to compare a project against a cost of capital rather than against another project, which is the question capital budgeting usually starts from. A rate is easy to talk about — this project returns 22 percent, our money costs 8 — and it does not depend on knowing how much you have to invest, so it works as a first screen on a long list of candidates. Three things it cannot do. It says nothing about size: a project that returns 200 percent on a 1,000 investment and one that returns 25 percent on a million can have the same internal rate of return, and only the net present value tells you which one to prefer. It assumes every receipt is reinvested at that same rate until the end, which is a strong assumption on a very high rate — the arithmetic is right, the story attached to it is optimistic. And it needs the cash flows to change sign only once: an outlay, then receipts. When a later period has a cost in it, the equation can have more than one solution, and this page refuses that input rather than silently returning one of the roots.

1. 100,000 invested, 205,000 returned over six years, money costing 8 percent

   1. The receipts total 205,000 against a cost of 100,000, but that comparison ignores when the money arrives
   2. Discounting each year at 8 percent gives a present value of 153,968.13 against the 100,000 cost, so the net present value is 53,968.13
   3. Solving for the rate that makes the net present value zero gives 22.49 percent
   4. 22.49 is above the 8 percent hurdle, so the verdict is accept

   Read the two answers together. The internal rate of return says the project can carry a cost of capital up to 22.49 percent, and the net present value says that at 8 percent it is worth 53,968.13 more than it costs — the same judgement, one expressed as a rate and one as money. The third figure, 205,000, is there to be subtracted from the investment by anyone who wants to skip the discounting; it is 105,000 more than the cost, and at a hurdle of zero the net present value would be exactly that.

2. A single receipt: 10,000 in, 12,000 back one period later

   1. With one period there is a closed form: 12,000 ÷ 10,000 − 1 = 20 percent
   2. Discounting the 12,000 at 20 percent gives 10,000, which is exactly the cost
   3. Net present value: 10,000 − 10,000 = 0
   4. The verdict is break-even

   This is the case the page checks its own solver against, because the answer can be worked out by hand. It is also the one place where a verdict can look strange: the net present value displays as 0.00 and the badge says break-even rather than reject. Anything that rounds to zero at two decimals is reported as break-even, because a reject verdict printed next to a net present value of 0.00 reads like an arithmetic error — and the display is what the reader checks the badge against.

3. The same project judged against a 25 percent hurdle

   1. The internal rate of return is unchanged at 20 percent — it does not depend on the hurdle
   2. Discounting 12,000 at 25 percent gives 9,600
   3. Net present value: 9,600 − 10,000 = −400
   4. 20 percent is below the 25 percent hurdle, so the verdict flips to reject

   Two runs of the same project with the same first output and opposite verdicts, which is the clearest way to see what each figure is for. The internal rate of return describes the project; the hurdle describes you. Change one and the verdict can flip without anything about the project having changed.

4. A project that destroys value: 10,000 in, 5,000 back

   1. One period again, so the closed form applies: 5,000 ÷ 10,000 − 1 = −50 percent
   2. Even with no discounting at all the receipt is 5,000 short of the cost
   3. Net present value at a hurdle of zero: 5,000 − 10,000 = −5,000
   4. The verdict is reject, and the negative rate is the honest description

   A negative internal rate of return is a real answer, not an error: it says the project would only break even if it could be funded at a negative cost of capital, which is to say it cannot. This is the case that decides the solver's lower bound — a version of this page borrowing a bond yield solver's range of zero to one hundred percent would throw an error here instead of returning minus 50.

5. Ten thousand in, one unit back after five years

   1. Zeros in the first four periods are legal: no cash arrives, but nothing is spent either
   2. The equation becomes 1 ÷ (1+r)^5 = 10,000 ÷ 10,000, so (1+r)^5 = 0.0001
   3. Solving gives r = −84.15 percent, not −100 percent, because the receipt is small but not zero
   4. Net present value at a hurdle of zero: 1 − 10,000 = −9,999

   Legitimate inputs can push the answer a long way below zero, and the page has to reach them or it will report a failure on a set of numbers that is perfectly well defined. The distinction worth keeping: this project has an internal rate of return of minus 84 percent, while a project that returns nothing at all has no internal rate of return, because no rate can make the value of nothing equal to a cost.

The internal rate of return is a rate, and rates hide size. Two projects can share a rate while differing by a factor of a thousand in the money at stake, so this figure is a screen rather than a decision: once a shortlist clears the hurdle, the net present value is what ranks them. It also assumes the receipts are reinvested at the internal rate of return itself until the end of the project, which is a fair description when the rate is modest and a stretch when it is 200 percent — the arithmetic still holds, but the interpretation that you will earn that rate on every dollar in the meantime does not. Cash flows must change sign exactly once, an outlay followed by receipts; a project with a cleanup cost in its final year can have two mathematically valid rates, and this page refuses that input rather than picking one. The rate is not adjusted for risk, taxes or inflation, and the discount rate you supply carries all of those assumptions by itself.

What is the internal rate of return?

It is the discount rate at which a project's future cash flows are worth exactly what the project costs — the rate at which the net present value is zero. Above that rate the project creates value; below it, the project destroys value. On the default numbers, investing 100,000 and receiving 205,000 over six years gives an internal rate of return of 22.49 percent, which means the project can carry a cost of capital of up to 22.49 percent before it stops being worth doing.

How is this different from the net present value?

They are the same equation with different unknowns. Net present value takes the discount rate as given and tells you what the project is worth; the internal rate of return takes the value as given, at zero, and tells you which discount rate achieves it. Because they are one equation, they can never disagree: a rate above your hurdle and a positive net present value are the same statement. What differs is what you have to know before you can run them — a net present value needs your cost of capital, while a rate can be quoted without it, which is why rates get compared across projects and net present values get used to rank them.

What should I use as the hurdle rate?

Whatever money genuinely costs you for this project: a borrowing rate if it is debt-financed, the return you would otherwise expect from the same money at similar risk if it is not, or a published rate where a rule requires one. The page will not choose it for you, and it is the single input most likely to be arbitrary. Its role is easy to see by running the same project twice: at a 20 percent hurdle a project returning 20 percent breaks even, and at 25 percent the identical project is rejected. Only the bar moved.

Why is the internal rate of return not solved with a formula?

Because there is no formula. A single receipt has a closed form — 12,000 back on 10,000 is 20 percent — but from two periods onwards the equation is a polynomial of that degree, and no rearrangement isolates the rate. The calculator instead narrows an interval: it finds a rate where the net present value is positive and one where it is negative, then repeatedly halves the gap between them until the value is zero to within two decimal places. That is also why the page is careful about the range it searches, since a project returning minus 84 percent has to fall inside it.

Why can the cash flows not contain a negative number?

Because the answer would stop being unique. With an outlay followed by receipts, the net present value falls steadily as the rate rises, so there is exactly one rate at which it crosses zero, and searching for it is guaranteed to find it. Put a cost in a later period — a decommissioning bill in year five — and the curve can bend back and cross zero twice, giving two mathematically valid internal rates of return. The page refuses negative cash flows and says so, rather than returning one of the two roots with nothing on screen to indicate that the other exists.

Can the internal rate of return be negative?

Yes, and it means the project does not return the money it consumes. Investing 10,000 and receiving 5,000 back gives minus 50 percent, which is to say the project would only break even if money cost less than nothing. The page deliberately searches below zero for this reason: an implementation that borrowed a range starting at zero, as a bond yield solver reasonably does, would report an error on this perfectly ordinary set of numbers.

When is the internal rate of return the wrong tool?

When the projects differ in size or in duration, and when the answer depends on what you do with the money in between. A rate cannot see scale, so a small project with a spectacular rate can outrank a large project that creates far more value in money, which is why capital budgeting pairs the rate with the net present value rather than using it alone. It also quietly assumes the receipts are reinvested at the same rate until the end, which is a reasonable description at 8 percent and a hopeful one at 200. Where either of those matters, rank by net present value and use the rate as a screen.

• Circular A-94, Guidelines and Discount Rates for Benefit-Cost Analysis of Federal Programs (revised November 9, 2023) — section 5a on the decision rule this page's verdict follows: projects with positive discounted net benefits generally increase welfare the most and are preferred, while those with negative discounted net benefits should generally be avoided — Office of Management and Budget, Executive Office of the President (United States)
• Compound Interest — the glossary definition of the growth that discounting runs backwards, which is the whole of the arithmetic behind a present value — Investor.gov, U.S. Securities and Exchange Commission (United States)